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1. Vector-Valued Maximal Inequalities and Multiparameter Oscillation Inequalities for the Polynomial Ergodic Averages Along Multi-dimensional Subsets of Primes

   https://doi.org/10.1007/s00041-024-10119-6  

   Mehlhop, Nathan (December 2024, Journal of Fourier Analysis and Applications)

   Abstract We prove uniform$$\ell ^2$$ ${\ell }^2$-valued maximal inequalities for polynomial ergodic averages and truncated singular operators of Cotlar type modeled over multidimensional subsets of primes. In the averages case, we combine this with earlier one-parameter oscillation estimates (Mehlhop and Słomian in Math Ann, 2023,https://doi.org/10.1007/s00208-023-02597-8) to prove corresponding multiparameter oscillation estimates. This provides a fuller quantitative description of the pointwise convergence of the mentioned averages and is a generalization of the polynomial Dunford–Zygmund ergodic theorem attributed to Bourgain (Mirek et al. in Rev Mat Iberoam 38:2249–2284, 2022). 

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2. An almost linear time algorithm testing whether the Markoff graph modulo p is connected

   https://doi.org/10.1007/s40993-024-00592-9  

   Brown, Colby Austin (March 2025, Research in Number Theory)

   Abstract The Markoff graphs modulopwere proven by Chen (Ann Math 199(1), 2024) to be connected for all but finitely many primes, and Baragar (The Markoff equation and equations of Hurwitz. Brown University, 1991) conjectured that they are connected for all primes, equivalently that every solution to the Markoff equation moduloplifts to a solution over$$\mathbb {Z}$$ $Z$. In this paper, we provide an algorithmic realization of the process introduced by Bourgain et al. [arXiv:1607.01530] to test whether the Markoff graph modulopis connected for arbitrary primes. Our algorithm runs in$$o(p^{1 + \epsilon })$$ $o\left(p^{1+ϵ}\right)$time for every$$\epsilon > 0$$ $ϵ>0$. We demonstrate this algorithm by confirming that the Markoff graph modulopis connected for all primes less than one million. 

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3. On a multi-parameter variant of the Bellow–Furstenberg problem

   https://doi.org/10.1017/fmp.2023.21  

   Bourgain, Jean; Mirek, Mariusz; Stein, Elias M.; Wright, James (January 2023, Forum of Mathematics, Pi)

   Abstract We prove convergence in norm and pointwise almost everywhere on$$L^p$$,$$p\in (1,\infty )$$, for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our result, in particular, gives an affirmative answer to a multi-parameter variant of the Bellow–Furstenberg problem. This paper is also the first systematic treatment of multi-parameter oscillation semi-norms which allows an efficient handling of multi-parameter pointwise convergence problems with arithmetic features. The methods of proof of our main result develop estimates for multi-parameter exponential sums, as well as introduce new ideas from the so-called multi-parameter circle method in the context of the geometry of backwards Newton diagrams that are dictated by the shape of the polynomials defining our ergodic averages. 

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4. Refined Chabauty–Kim computations for the thrice-punctured line over $$\mathbb {Z}[1/6]$$

   https://doi.org/10.1007/s40993-024-00597-4  

   Lüdtke, Martin (March 2025, Research in Number Theory)

   Abstract The Chabauty–Kim method and its refined variant by Betts and Dogra aim to cut out theS-integral points$$X(\mathbb {Z}_S)$$ $X\left(Z_S\right)$on a curve inside thep-adic points$$X(\mathbb {Z}_p)$$ $X\left(Z_p\right)$by producing enough Coleman functions vanishing on them. We derive new functions in the case of the thrice-punctured line whenScontains two primes. We describe an algorithm for computing refined Chabauty–Kim loci and verify Kim’s Conjecture over$$\mathbb {Z}[1/6]$$ $Z[1/6]$for all choices of auxiliary prime $$p < 10{,}000$$ $p<10,000$. 

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5. Hypergeometric L-functions in average polynomial time, II

   https://doi.org/10.1007/s40993-024-00593-8  

   Costa, Edgar; Kedlaya, Kiran S; Roe, David (January 2025, Research in Number Theory)

   Abstract We describe an algorithm for computing, for all primes$$p \le X$$ $p\le X$, the trace of Frobenius atpof a hypergeometric motive over$$\mathbb {Q}$$ $Q$in time quasilinear inX. This involves computing the trace modulo$$p^e$$ $p^e$for suitablee; as in our previous work treating the case$$e=1$$ $e=1$, we combine the Beukers–Cohen–Mellit trace formula with average polynomial time techniques of Harvey and Harvey–Sutherland. The key new ingredient for$$e>1$$ $e>1$is an expanded version of Harvey’s “generic prime” construction, making it possible to incorporate certainp-adic transcendental functions into the computation; one of these is thep-adic Gamma function, whose average polynomial time computation is an intermediate step which may be of independent interest. We also provide an implementation in Sage and discuss the remaining computational issues around tabulating hypergeometricL-series. 

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